| L(s) = 1 | − 1.61·2-s − 1.23·3-s + 0.618·4-s + 2.00·6-s − 7-s + 2.23·8-s − 1.47·9-s + 4.23·11-s − 0.763·12-s + 3.23·13-s + 1.61·14-s − 4.85·16-s + 6.47·17-s + 2.38·18-s + 4.47·19-s + 1.23·21-s − 6.85·22-s − 1.76·23-s − 2.76·24-s − 5.23·26-s + 5.52·27-s − 0.618·28-s + 5·29-s − 9.70·31-s + 3.38·32-s − 5.23·33-s − 10.4·34-s + ⋯ |
| L(s) = 1 | − 1.14·2-s − 0.713·3-s + 0.309·4-s + 0.816·6-s − 0.377·7-s + 0.790·8-s − 0.490·9-s + 1.27·11-s − 0.220·12-s + 0.897·13-s + 0.432·14-s − 1.21·16-s + 1.56·17-s + 0.561·18-s + 1.02·19-s + 0.269·21-s − 1.46·22-s − 0.367·23-s − 0.564·24-s − 1.02·26-s + 1.06·27-s − 0.116·28-s + 0.928·29-s − 1.74·31-s + 0.597·32-s − 0.911·33-s − 1.79·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5073493884\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5073493884\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 3 | \( 1 + 1.23T + 3T^{2} \) |
| 11 | \( 1 - 4.23T + 11T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 17 | \( 1 - 6.47T + 17T^{2} \) |
| 19 | \( 1 - 4.47T + 19T^{2} \) |
| 23 | \( 1 + 1.76T + 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 + 9.70T + 31T^{2} \) |
| 37 | \( 1 + 3T + 37T^{2} \) |
| 41 | \( 1 - 9.23T + 41T^{2} \) |
| 43 | \( 1 + 6.23T + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 - 0.472T + 53T^{2} \) |
| 59 | \( 1 + 1.70T + 59T^{2} \) |
| 61 | \( 1 - 3.70T + 61T^{2} \) |
| 67 | \( 1 + 0.236T + 67T^{2} \) |
| 71 | \( 1 + 4.70T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 5.70T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + 0.763T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.36216294739086119519615816013, −11.53100478515449172831423995482, −10.62542567245760530310988346609, −9.615821546120246046133646690672, −8.841567132457867444396120965366, −7.72279593523330815772616751509, −6.50672188460512835567491340138, −5.39509543543337615067873355224, −3.64739169163803135623810162905, −1.09895466840553961393358200917,
1.09895466840553961393358200917, 3.64739169163803135623810162905, 5.39509543543337615067873355224, 6.50672188460512835567491340138, 7.72279593523330815772616751509, 8.841567132457867444396120965366, 9.615821546120246046133646690672, 10.62542567245760530310988346609, 11.53100478515449172831423995482, 12.36216294739086119519615816013
